Nonlinear Sciences – Exactly Solvable and Integrable Systems
Scientific paper
2012-04-10
Nonlinear Sciences
Exactly Solvable and Integrable Systems
57 pages. arXiv admin note: substantial text overlap with arXiv:solv-int/9809004
Scientific paper
Though completely integrable Camassa-Holm (CH) equation and Degasperis-Procesi (DP) equation are cast in the same peakon family, they possess the second- and third-order Lax operators, respectively. From the viewpoint of algebro-geometrical study, this difference lies in hyper-elliptic and non-hyper-elliptic curves. The non-hyper-elliptic curves lead to great difficulty in the construction of algebro-geometric solutions of the DP equation. In this paper, we derive the DP hierarchy with the help of Lenard recursion operators. Based on the characteristic polynomial of a Lax matrix for the DP hierarchy, we introduce a third order algebraic curve $\mathcal{K}_{r-2}$ with genus $r-2$, from which the associated Baker-Ahhiezer functions, meromorphic function and Dubrovin-type equations are established. Furthermore, the theory of algebraic curve is applied to derive explicit representations of the theta function for the Baker-Ahhiezer functions and the meromorphic function. In particular, globally algebro-geometric solutions are obtained for all equations in the whole DP hierarchy.
Fan Engui
Hou Yu
Qiao Zhijun
Zhao Peng
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